Question

61–64. Polar equations for conic sections Graph the following conic sections, labeling vertices, foci, directrices, and asymptotes (if they exist). Give the eccentricity of the curve. Use a graphing utility to check your work.

r = 3/(1 - 2 cos θ)

Accepted Answer

Identify the form of the polar equation. The given equation is $$r = \frac{3}{1 - 2 \cos 	heta}$$, which matches the standard form for conic sections in polar coordinates: $$r = \frac{ed}{1 + e \cos 	heta} or$$ $$r = \frac{ed}{1 + e \sin 	heta}$$, where $$e is $$the eccentricity and $$d is $$the distance from the focus to the directrix. Rewrite the equation to match the standard form $$r = \frac{ed}{1 + e \cos 	heta}. $$Notice the denominator is $$1 - 2 \cos 	heta$$, so we can write it as $$1 + (-2) \cos 	heta. $$This means $$e = 2$$ and $$ed = 3$$, so $$d = \frac{3}{e} = \frac{3}{2}. $$Determine the type of conic based on the eccentricity $$e. $$Since $$e = 2 \u003e 1$$, the conic is a hyperbola. Find the vertices by analyzing the values of $$r$$ when the denominator is minimized or maximized. The vertices occur at angles where $$1 - 2 \cos 	heta is $$smallest (which makes $$r$$ largest) and where it is largest (which makes $$r$$ smallest). Calculate $$r at $$these key angles, such as $$	heta = 0$$ and $$	heta = \pi. $$Locate the focus at the pole (origin), find the directrix using $$d = \frac{3}{2}$$, and sketch the asymptotes of the hyperbola based on the behavior of $$r as$$ $$	heta$$ approaches values that make the denominator approach zero. Label the vertices, foci, directrices, and asymptotes on the graph.

61–64. Polar equations for conic sections Graph the following conic sections, labeling vertices, foci, directrices, and asymptotes (if they exist). Give the eccentricity of the curve. Use a graphing utility to check your work.

r = 3/(1 - 2 cos θ)

Verified video answer for a similar problem:

Key Concepts

Polar Form of Conic Sections

Eccentricity and Classification of Conics

Graphing and Identifying Key Features

61–64. Polar equations for conic sections Graph the following conic sections, labeling vertices, foci, directrices, and asymptotes (if they exist). Give the eccentricity of the curve. Use a graphing utility to check your work.r = 3/(1 - 2 cos θ)

Verified video answer for a similar problem:

Key Concepts

Polar Form of Conic Sections

Eccentricity and Classification of Conics

Graphing and Identifying Key Features

61–64. Polar equations for conic sections Graph the following conic sections, labeling vertices, foci, directrices, and asymptotes (if they exist). Give the eccentricity of the curve. Use a graphing utility to check your work.

r = 3/(1 - 2 cos θ)